### Related Formula S_infty = fraca1 - r$S_{\infty} = \frac{a}{1 - r}$ Where S_infty$S_{\infty}$ is the sum of an infinite geometric progression, a$a$ is the first term, and r$r$ is the common ratio. ### Core Logic Let success (S$S$) be rolling a 2, and failure (F$F$) be rolling anything else. P(S) = frac16$P(S) = \frac{1}{6}$ P(F) = frac56$P(F) = \frac{5}{6}$ We want the probability that the first success occurs on an even number of throws (2nd, 4th, 6th, dots$\dots$). The sequence of events for success on even throws is: - Success on 2nd throw: F, S$F, S$ - Success on 4th throw: F, F, F, S$F, F, F, S$ - Success on 6th throw: F, F, F, F, F, S$F, F, F, F, F, S$ Writing this as a sum of probabilities: textRequired Probability = P(F)P(S) + P(F)^3 P(S) + P(F)^5 P(S) + dots$\text{Required Probability} = P(F)P(S) + P(F)^3 P(S) + P(F)^5 P(S) + \dots$ = left(frac56right)left(frac16right) + left(frac56right)^3left(frac16right) + left(frac56right)^5left(frac16right) + dots$= \left(\frac{5}{6}\right)\left(\frac{1}{6}\right) + \left(\frac{5}{6}\right)^3\left(\frac{1}{6}\right) + \left(\frac{5}{6}\right)^5\left(\frac{1}{6}\right) + \dots$ ### Step 1: Compute Infinite Series Sum This is an infinite geometric series with: First term a = frac56 times frac16 = frac536$a = \frac{5}{6} \times \frac{1}{6} = \frac{5}{36}$ Common ratio r = left(frac56right)^2 = frac2536$r = \left(\frac{5}{6}\right)^2 = \frac{25}{36}$ Applying the sum formula: S_infty = fracfrac5361 - frac2536$S_{\infty} = \frac{\frac{5}{36}}{1 - \frac{25}{36}}$ = fracfrac536frac1136$= \frac{\frac{5}{36}}{\frac{11}{36}}$ = frac511$= \frac{5}{11}$ ### Pattern Recognition For alternating success/failure probabilities P(textEven) = fracq cdot p1 - q^2$P(\text{Even}) = \frac{q \cdot p}{1 - q^2}$ and P(textOdd) = fracp1 - q^2$P(\text{Odd}) = \frac{p}{1 - q^2}$. Knowing this format immediately turns it into a 5-second mental calculation: frac(5/6)(1/6)1 - 25/36 = 5/11$\frac{(5/6)(1/6)}{1 - 25/36} = 5/11$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Probability Class 11 Mathematics: Sequences and Series